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Zero-Divisor Graphs and Lattices of Finite Commutative Rings

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Zero-Divisor Graphs and Lattices of Finite Commutative Rings Rose-Hulman Undergraduate Mathematics Journal Volume 12 Issue 1 Article 4 Zero-Divisor Graphs and Lattices of Finite Commutative Rings Darrin Weber Millikin University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Weber, Darrin (2011) "Zero-Divisor Graphs and Lattices of Finite Commutative Rings," Rose-Hulman Undergraduate Mathematics Journal: Vol. 12 : Iss. 1 , Article 4. Available at: https://scholar.rose-hulman.edu/rhumj/vol12/iss1/4 Rose- Hulman Undergraduate Mathematics Journal Zero-Divisor Graphs and Lattices of Finite Commutative Rings Darrin Webera Volume 12, no. 1, Spring 2011 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal Millikin University, Decatur, IL, [email protected] Rose-Hulman Undergraduate Mathematics Journal Volume 12, no. 1, Spring 2011 Zero-Divisor Graphs and Lattices of Finite Commutative Rings Darrin Weber Abstract.In this paper we consider, for a finite commutative ring R, the well- studied zero-divisor graph Γ(R) and the compressed zero-divisor graph Γc(R) of R and a newly-defined graphical structure — the zero-divisor lattice Λ(R) of R. We give results which provide information when Γ(R) =∼ Γ(S), Γc(R) =∼ Γc(S), and Λ(R) =∼ Λ(S) for two finite commutative rings R and S. We also provide a theorem which says that Λ(R) is almost always connected. Acknowledgements: This paper is the result of a summer’s worth of undergraduate research at Millikin University and was funded by Millikin University’s Student Undergrad- uate Research Fund. The author would like to thank Dr. Joe Stickles for his guidance and advice throughout the development of this paper, Dr. Michael Axtell for his comments and suggestions, and Dr. James Rauff for his helpful input. Page58 RHITUndergrad.Math.J.,Vol. 12, no. 1 1 Introduction The notion of a zero-divisor graph was introduced in [7] and was studied further in [4]. However, the definition we will provide of the zero-divisor graph of a ring is more generally accepted and was first used in [3]. Much research has been done on zero-divisor graphs over the past ten years, and many of the papers can be found in the reference section of [2]. The hope is that the graph-theoretic properties of the zero-divisior graph will help us better understand the ring-theoretic properties of R. Not only have zero-divisor graphs been studied by professional mathematicians, but they have also been the focus of master’s theses and doctorial dissertations. Further, many under- graduates have researched zero-divisor graphs extensively providing a number of substantial results in the field. In particular, the Wabash Summer Institute in Mathematics at Wabash College has, in part, focused on the interplay between ring structure and zero-divisor graph structure. Some tools used to aid in the research process in this program were Mathematica notebooks that displayed the zero-divisor graph of certain rings. These notebooks have been rewritten as well as additional notebooks added, and they can be found at [14]. All of the graphs displayed in this paper were generated using those notebooks. To help study large zero-divisor graphs, we introduce another related definition to the zero-divisor graph, called the compressed zero-divisor graph, which first appeared in a similar form in [12] and was further studied in [13]. We also introduce the definition for the zero- divisor lattice suggested by Dr. Nicholas Baeth of the University of Central Missouri. In section 2, we provide the necessary definitions for this paper. In section 3, we look at the connections between the zero-divisor graph and the compressed zero-divisor graph and prove that isomorphic graphs yield isomorphic compressed graphs in Theorem 3.1. In Theorem 3.3, we also demonstrate that in most cases cut-sets are preserved when looking at the compressed zero-divisor graph. In section 4, we explore the connections between all three graphical structures and show in Theorems 4.1 and 4.2 that isomorphic graphs or isomorphic compressed graphs gives us isomorphic lattices. In section 5, we show that the zero-divisor lattice is connected in almost every case. 2 Definitions Throughout this paper, R denotes a finite commutative ring with identity. An element a is a zero-divisor if there exists a nonzero r ∈ R such that ar = 0. We denote the set of all zero- divisors in R as Z(R). The set of annihilators of a ring element x is ann(x)= {a | ax =0}. A ring is called local if it has one unique maximal ideal. (A maximal ideal is an ideal A of a ring R such that if A ⊆ B ⊆ R, where B is also an ideal, then either A = B or B = R.) A field is a commutative ring with identity in which every nonzero element has a multiplicative inverse. For a graph G, we denote the set of vertices of G as V (G) and the set of edges as E(G). We define a path between two elements a1,an ∈ V (G) to be an ordered sequence of distinct vertices and edges {a1,e1,a2,...,en−1,an} of G such that edge ei, denoted by ai RHIT Undergrad. Math. J., Vol. 12, no. 1 Page 59 — ai+1, is incident to vertices ai and ai+1, for each i ∈ {1,...,n − 1}. For x,y ∈ V (G), the minimum length of all paths from x to y, if it exists, is called the distance from x to y and is denoted d(x,y). If no path from x to y exists, then d(x,y) = ∞. The diameter of a graph is diam(G) = sup{d(x,y) | x,y ∈ V (G)}. The neighborhood of a vertex is the set nbd(x) = {z ∈ V (G) | x — z}. A graph is connected if a path exists between any two distinct vertices. A complete r-partite graph is the disjoint union of r nonempty vertex sets in which two distinct vertices are adjacent if and only if they are in distinct vertex sets. In the case where r = 2, we call the graph complete bipartite. Two graphs G and H are said to be isomorphic if there exists a one-to-one and onto function φ : V (G) → V (H) such that if x,y ∈ V (G), then x — y if and only if φ(x)— φ(y). Lemma 2.1 shows that graph isomorphism preserves neighborhoods. We provide a proof for completeness. Lemma 2.1. Let G ∼= H. If φ(x)= y, then φ(nbd(x)) = nbd(y). Proof. Let φ : G → H be a graph isomorphism. Let x ∈ V (G) and φ(x)= y ∈ V (H). Then φ(nbd(x)) = {φ(z) | x — z} = {φ(z) | φ(x) — φ(z)} = {φ(z) | y — φ(z)} = nbd(y). The zero-divisor graph of a ring R, denoted Γ(R), is a graph with V (Γ(R)) = Z(R)\{0} and E(Γ(R)) = {a — b | ab = 0}. By [3], we know that Γ(R) is always connected and diam(Γ(R)) ≤ 3 for any ring R. Notice that nbd(x) = ann(x) in a zero-divisor graph. A cut-set of a graph G is a set A ⊂ V (G) minimal among all subsets of V (G) such that there exist distinct vertices c,d ∈ V (G)\A such that every path from c to d involves at least one element of A. Example 2.2. In Γ(Z30), shown in Figure 1(a), there are three cut-sets: {15}, {10, 20}, and {6, 12, 18, 24}. The cut-sets in Γc(R) are {15}, {10}, and {6} and are shown in Figure 1(b). A cut vertex is a cut-set of size 1. The study of cut vertices of zero-divisor graphs began in [6] and was continued in [9]. In [6], it was shown that a cut vertex along with zero form an ideal in the ring. In [9], the cut vertex was generalized to cut-sets, and cut-sets were classified for finite, nonlocal commutative rings. In addition, it was shown that the cut-set along with zero form an ideal in the ring. For algebraic definitions and concepts not listed here, see [10], and for graph theory definitions and concepts, see [8]. To define both the compressed zero-divisor graph and the zero-divisor lattice, we first need to define an equivalence relation on the zero-divisors of R. Definition 2.1. Let R be a commutative ring. Define a relation ≡ on R by x ≡ y if and only if ann(x) = ann(y). It is easy to see that ≡ is an equivalence relation on R. We denote the equivalence class of x byx ¯. Notice that ann(0) = R andx ¯ =y ¯ for all x,y ∈ R\Z(R). However, we will only focus on the equivalence classes of the nonzero zero-divisors. Page60 RHITUndergrad.Math.J.,Vol. 12, no. 1 Definition 2.2. For a ring R, the compressed zero-divisor graph, denoted Γc(R), is a graph whose vertices are the equivalence classes of the nonzero zero-divisors, and two vertices a¯ and ¯b are connected by an edge if and only if ab =0. Example 2.3. For Z30, Figure 1 shows the difference between the zero-divisor graph, (a), and the compressed zero-divisor graph, (b). 2 3 2 3 4 9 8 15 10 21 10 14 20 27 15 16 6 6 12 22 18 24 26 28 5 25 5 (a) Γ(Z30) (b) Γc(Z30) Figure 1: The zero-divisor graph and compressed zero-divisor graph of Z30 Notice that by Theorem 2.8 in [3], when the zero-divisor graph is a complete graph, either the ring is Z2 × Z2, or every vertex loops to itself.
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